Seminario del 2022

Settembre
20
2022
Giovanni Cupini
The regularity of solutions to elliptic systems in divergence form
nell'ambito della serie: TOPICS IN MATHEMATICS 2021/2022
analisi matematica
The theory of elliptic equations and systems of m equations in divergence form, is strictly related to the theory of minimization of integral functionals. After a review on the existence issue, we will focus on the regularity problem: under which conditions the solutions are regular? The ideal process to prove that a (weak) solution, apriori only in a Sobolev space W^{1,p}, is C^{\infty} will be sketched. Unfortunately, the gain in regularity is not for free, and it is guaranteed only if particular conditions are met. In the past years, counterexamples have shown that: 1) under certain growth conditions the regularity can be lost, even in the scalar case; 2) in the vectorial case the situation is far worse, since even solutions to linear and uniformly elliptic systems may be locally unbounded (!). The main effort is to find conditions that force the regularity of the solutions. We will focus in particular to the vectorial case; i.e. the local regularity of weak solutions to elliptic systems. The main and most common structure condition, that forces, in general, regularity in the vectorial setting, is the so called Uhlenbeck’s structure (dependence on the modulus of the gradient). Meier, in 1982, introduced another assumption, related to a so called Indicator function: a more general condition than Uhlenbeck’s one, that allows to include more general systems. For them, Meier proved the local boundedness of the solutions. We will exhibit examples of systems that do not satisfy the Meier’s condition, but for which, in a recent result in collaboration with F. Leonetti (L’Aquila) and E. Mascolo (Firenze), we proved the boundedness of the solutions. The crucial structure assumption is the componentwise coercivity introduced by Bjorn in 2001.
organizzato da: Antonella Grassi e Stefano Pagliarani

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